Inverse Problem Solution and Regularization Parameter Selection for Current Distribution Reconstruction in Switching Arcs by Inverting Magnetic Fields

<div><i>Simulation I</i>. Plot of <svg height="11.5564pt" id="M169" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 25.4664 11.5564" width="25.4664pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path></g><g transform="matrix(.013,0,0,-0.013,8.892,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,13.39,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z" id="g113-223"></path></g><g transform="matrix(.013,0,0,-0.013,20.776,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g></svg> versus <svg height="6.1673pt" id="M170" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 7.51131 6.1673" width="7.51131pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z" id="g113-223"></path></g></svg> (a), plot of <svg height="15.6419pt" id="M171" style="vertical-align:-3.3837pt" version="1.1" viewbox="-0.0498162 -12.2582 104.734 15.6419" width="104.734pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M238 0V26C174 32 166 38 166 104V712C132 700 70 683 18 677V653C81 647 87 645 87 577V104C87 38 78 32 15 26V0H238Z" id="g121-106"></path></g><g transform="matrix(.013,0,0,-0.013,3.289,0)"><path d="M257 449C165 449 37 374 37 209C37 98 119 -12 256 -12C355 -12 473 65 473 226C473 349 381 449 257 449ZM244 416C333 416 380 320 380 204C380 67 329 21 267 21C184 21 130 115 130 241C130 354 184 416 244 416Z" id="g121-109"></path></g><g transform="matrix(.013,0,0,-0.013,9.919,0)"><path d="M463 437C426 431 375 425 327 422C297 440 264 449 231 449H230C153 449 51 396 51 283C51 215 94 168 139 149C123 129 91 103 51 88C50 78 53 60 62 46C75 25 100 2 136 -9C112 -28 73 -59 53 -79C38 -94 29 -113 29 -135C30 -192 91 -257 203 -257C336 -257 452 -160 452 -59C452 39 371 59 309 59C275 59 240 58 203 58C158 58 140 77 140 96C140 110 157 129 170 138C186 135 205 133 221 133C306 133 396 185 396 293C396 328 384 360 366 381L423 378C439 387 459 413 468 429L463 437ZM219 418C277 418 314 362 314 284C314 205 275 166 231 165C176 165 137 221 137 299C137 376 177 418 219 418ZM241 -11C285 -11 314 -14 339 -24C367 -36 384 -61 384 -95C384 -157 335 -206 240 -206C166 -206 108 -165 108 -110C108 -82 128 -54 154 -32C172 -17 195 -11 241 -11Z" id="g121-101"></path></g><g transform="matrix(.013,0,0,-0.013,18.242,0)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.013,0,0,-0.013,23.585,0)"><path d="M969 650H734V615C815 608 819 591 805 531C791 472 750 316 712 193H709C654 339 597 492 541 644H493C447 490 397 339 349 197H347C313 307 279 424 246 541C232 592 233 606 308 615V650H13V615C72 607 89 593 113 515C169 337 218 169 273 -11H325C373 133 424 282 475 433H477C533 281 586 138 641 -11H693C777 269 836 443 858 508C886 591 902 605 969 615V650Z" id="g13-85"></path></g><g transform="matrix(.013,0,0,-0.013,36.312,0)"><path d="M337 650H39V615C115 608 125 602 125 517V190C125 116 123 51 114 -4C104 -68 74 -103 20 -126L35 -162C75 -153 117 -134 157 -100C249 -25 259 70 259 186V517C259 602 267 608 337 615V650Z" id="g13-72"></path></g><g transform="matrix(.0091,0,0,-0.0091,41.136,3.132)"><path d="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z" id="g50-223"></path></g><g transform="matrix(.0091,0,0,-0.0091,46.95,-5.741)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" id="g50-41"></path></g><g transform="matrix(.0091,0,0,-0.0091,50.208,-5.741)"><path d="M501 517C491 587 426 710 310 710C240 710 175 662 175 601C175 561 201 517 250 452C219 438 187 423 157 404C88 360 24 280 24 178C24 71 90 -12 195 -12C250 -12 297 6 334 32C412 87 452 170 452 249C452 335 408 396 333 481C259 565 229 601 229 621C229 642 245 652 275 652C359 652 427 583 490 499L501 517ZM364 233C364 142 323 33 222 33C173 33 119 81 119 178C119 265 156 335 193 374C213 395 242 413 272 425C308 384 364 319 364 233Z" id="g50-226"></path></g><g transform="matrix(.0091,0,0,-0.0091,54.986,-5.741)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z" id="g50-42"></path></g><g transform="matrix(.013,0,0,-0.013,61.689,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path></g><g transform="matrix(.013,0,0,-0.013,72.225,0)"><path d="M746 650H445V615C520 608 528 602 528 517V370H255V517C255 602 263 608 335 615V650H35V615C112 608 121 602 121 517V133C121 48 112 42 39 35V0H337V35C263 42 255 48 255 133V319H528V133C528 48 520 42 441 35V0H746V35C670 42 662 48 662 132V517C662 602 668 608 746 615V650Z" id="g13-70"></path></g><g transform="matrix(.0091,0,0,-0.0091,82.404,-5.741)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" id="g50-41"></path></g><g transform="matrix(.0091,0,0,-0.0091,85.662,-5.741)"><path d="M501 517C491 587 426 710 310 710C240 710 175 662 175 601C175 561 201 517 250 452C219 438 187 423 157 404C88 360 24 280 24 178C24 71 90 -12 195 -12C250 -12 297 6 334 32C412 87 452 170 452 249C452 335 408 396 333 481C259 565 229 601 229 621C229 642 245 652 275 652C359 652 427 583 490 499L501 517ZM364 233C364 142 323 33 222 33C173 33 119 81 119 178C119 265 156 335 193 374C213 395 242 413 272 425C308 384 364 319 364 233Z" id="g50-226"></path></g><g transform="matrix(.0091,0,0,-0.0091,90.44,-5.741)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z" id="g50-42"></path></g><g transform="matrix(.013,0,0,-0.013,94.238,0)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.0091,0,0,-0.0091,99.581,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg> versus <svg height="15.6419pt" id="M172" style="vertical-align:-3.3837pt" version="1.1" viewbox="-0.0498162 -12.2582 63.6277 15.6419" width="63.6277pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M238 0V26C174 32 166 38 166 104V712C132 700 70 683 18 677V653C81 647 87 645 87 577V104C87 38 78 32 15 26V0H238Z" id="g121-106"></path></g><g transform="matrix(.013,0,0,-0.013,3.289,0)"><path d="M257 449C165 449 37 374 37 209C37 98 119 -12 256 -12C355 -12 473 65 473 226C473 349 381 449 257 449ZM244 416C333 416 380 320 380 204C380 67 329 21 267 21C184 21 130 115 130 241C130 354 184 416 244 416Z" id="g121-109"></path></g><g transform="matrix(.013,0,0,-0.013,9.919,0)"><path d="M463 437C426 431 375 425 327 422C297 440 264 449 231 449H230C153 449 51 396 51 283C51 215 94 168 139 149C123 129 91 103 51 88C50 78 53 60 62 46C75 25 100 2 136 -9C112 -28 73 -59 53 -79C38 -94 29 -113 29 -135C30 -192 91 -257 203 -257C336 -257 452 -160 452 -59C452 39 371 59 309 59C275 59 240 58 203 58C158 58 140 77 140 96C140 110 157 129 170 138C186 135 205 133 221 133C306 133 396 185 396 293C396 328 384 360 366 381L423 378C439 387 459 413 468 429L463 437ZM219 418C277 418 314 362 314 284C314 205 275 166 231 165C176 165 137 221 137 299C137 376 177 418 219 418ZM241 -11C285 -11 314 -14 339 -24C367 -36 384 -61 384 -95C384 -157 335 -206 240 -206C166 -206 108 -165 108 -110C108 -82 128 -54 154 -32C172 -17 195 -11 241 -11Z" id="g121-101"></path></g><g transform="matrix(.013,0,0,-0.013,18.242,0)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.013,0,0,-0.013,23.585,0)"><path d="M512 180C494 129 475 97 457 77C433 50 397 43 354 43C323 43 297 45 279 52C259 61 255 78 255 125V516C255 602 264 608 340 615V650H35V615C111 608 121 602 121 516V133C121 48 111 42 24 35V0H511C519 27 540 126 548 169L512 180Z" id="g13-74"></path></g><g transform="matrix(.013,0,0,-0.013,30.748,0)"><path d="M337 650H39V615C115 608 125 602 125 517V190C125 116 123 51 114 -4C104 -68 74 -103 20 -126L35 -162C75 -153 117 -134 157 -100C249 -25 259 70 259 186V517C259 602 267 608 337 615V650Z" id="g13-72"></path></g><g transform="matrix(.0091,0,0,-0.0091,35.551,3.132)"><path d="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z" id="g50-223"></path></g><g transform="matrix(.0091,0,0,-0.0091,41.365,-5.741)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" id="g50-41"></path></g><g transform="matrix(.0091,0,0,-0.0091,44.623,-5.741)"><path d="M501 517C491 587 426 710 310 710C240 710 175 662 175 601C175 561 201 517 250 452C219 438 187 423 157 404C88 360 24 280 24 178C24 71 90 -12 195 -12C250 -12 297 6 334 32C412 87 452 170 452 249C452 335 408 396 333 481C259 565 229 601 229 621C229 642 245 652 275 652C359 652 427 583 490 499L501 517ZM364 233C364 142 323 33 222 33C173 33 119 81 119 178C119 265 156 335 193 374C213 395 242 413 272 425C308 384 364 319 364 233Z" id="g50-226"></path></g><g transform="matrix(.0091,0,0,-0.0091,49.401,-5.741)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z" id="g50-42"></path></g><g transform="matrix(.013,0,0,-0.013,53.198,0)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.0091,0,0,-0.0091,58.542,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg> (b), and the <svg height="11.5564pt" id="M173" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 25.4664 11.5564" width="25.4664pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path></g><g transform="matrix(.013,0,0,-0.013,8.892,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,13.39,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z" id="g113-223"></path></g><g transform="matrix(.013,0,0,-0.013,20.776,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g></svg> curve versus (c). The minimal points of <svg height="11.5564pt" id="M174" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 25.4664 11.5564" width="25.4664pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path></g><g transform="matrix(.013,0,0,-0.013,8.892,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,13.39,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z" id="g113-223"></path></g><g transform="matrix(.013,0,0,-0.013,20.776,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g></svg> and of <svg height="11.5564pt" id="M175" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 25.9622 11.5564" width="25.9622pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M699 368C699 549 574 666 407 666C186 666 23 488 23 277C23 113 129 -3 288 -13L307 -26C431 -111 501 -139 533 -147C559 -154 613 -163 658 -164L666 -141C597 -111 507 -66 430 -11L416 -1C580 42 699 190 699 368ZM601 371C601 227 518 54 381 22L354 40L278 24C175 47 120 145 120 269C120 451 235 631 398 631C540 631 601 521 601 371Z" id="g113-82"></path></g><g transform="matrix(.013,0,0,-0.013,9.386,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,13.884,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z" id="g113-223"></path></g><g transform="matrix(.013,0,0,-0.013,21.27,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g></svg> are marked out with the red asterisk.</div>

Mathematical Problems in Engineering

fig5

Figure 5

Figure 5: Inverse Problem Solution and Regularization Parameter Selection for Current Distribution Reconstruction in Switching Arcs by Inverting Magnetic Fields 

Figure 5 | Inverse Problem Solution and Regularization Parameter Selection for Current Distribution Reconstruction in Switching Arcs by Inverting Magnetic Fields 